where \(e(q,u)=(-\Delta )^{-1}\big (\nabla \cdot (q\nabla u)+f\big )\)  and \(e(\cdot , \cdot )\)  can be viewed as an operator from \(K\times H_0^1(\Omega )\)  to \(H_0^1(\Omega )\) .
In [1]}, it is proved that the discretized augmented Lagrangian functional (REF ) exists at least one saddle-point, and the finite element solution \((u_h,p_h)\)  converges to the solution of (REF ). In [1]}, [3]}, it is suggested to apply the classic augmented Lagrangian method (ALM) originally proposed in [4]}, [5]} to (REF ), and the iterative scheme reads as
\(\left\lbrace \begin{aligned}&(q_h^{k+1},u_h^{k+1})=\underset{(q_h,u_h)\in K_h\times \bar{V}_h}{\arg \min }~ {L}_{\gamma _k}(q_h,u_h;\mu _h^k),\\&\mu _h^{k+1}=\mu _h^k+\gamma _k e(q_h^{k+1},u_h^{k+1}).\\\end{aligned}\right.\) 
